Finite Group Databasebeta

← #28915

Subgroups — #28915

Order 256.

442 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic, complemented
4#2427C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic
7#2015C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1969C211yesmaximal, characteristic, complemented
10#1935C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#489—24no—
26#468—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#400—21no—
41#397—21no—
42#394—21no—
43#393—21no—
44#391—21no—
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#386—21no—
51#380—21no—
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×Q16—22no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83#124—22no—
84#124—41no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89#121—22no—
90#121—41no—
91C4×D4—22no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—41no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#102—22no—
122#102—22no—
123#102D414yes—
124C22⋊C8—22no—
125#97—22no—
126C4×C8—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—81no—
163C2×D4C2×D412yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2411yescharacteristic
186Q16—44no—
187SD16—44no—
188M16—44no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8C2×D412yes—
199C2×C8#4112yes—
200C2×C8#4112yes—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C4⋊C4#4112yes—
215C22⋊C4—22no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—81no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C22×D411yescharacteristic
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—84no—
253D4—22no—
254D4—22no—
255D4—41no—
256D4—41no—
257D4—84no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—84no—
298C2×C4#13712yes—
299C2×C4C22×D411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C8—22no—
306C8—22no—
307C8—22no—
308C8—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#53211yescharacteristic
324C22D4211yescharacteristic
325C22#54611yescharacteristic
326C22C2×#13711yescharacteristic
327C22#57511yescharacteristic
328C22#57611yescharacteristic
329C22#57811yescharacteristic
330C4—22no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—42no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#53211yescharacteristic
343C4#53811yescharacteristic
344C4#54611yescharacteristic
345C4#55511yescharacteristic
346C2—81no—
347C2—82no—
348C2#255511yescharacteristic
349C2#317211yescharacteristic
350C2#317511yescharacteristic
351C2#327311yescharacteristic
352C2#327611yescharacteristic
353C2#331311yescharacteristic
354C2#331411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#489C2semidirect
3. #2492C2semidirect
5. #2057C2semidirect
7. #2015C2semidirect
8. #2001C2semidirect
9. #1969C2semidirect
10. #1935C2semidirect
11. #1928C2semidirect
12. #1894C2semidirect
13. #1877C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#4892
C2×#4682
#24922
#24272
#20572
#20332
#20152
#20012
#19692
#19352
#19282
#18942
#18772
C4×#1022
#17532